Matrices conjugate transpose

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

Example 15.4. Determine the conjugate transpose of the complex matrix C ... 

The singular values of a complex matrix are similar to those of a reof matrix. The only difference is that we use the conjugate transpose. 

MADD adds complex matrices 4 nd to give C MMULT multiplies complex matrices and to give C IDENT forms an N x Af identity matrix I CONJT takes the conjugate transpose of matrix ... 

We denote the set of complex numbers by C. Readers should be familiar with complex numbers and how to add and multiply them, as described in many standard calculus texts. We use i to denote the square root of —1 and an asterisk to denote complex conjugation if x and y are real numbers, then (x + iy) = X — iy. Later in the text, we will use the asterisk to denote the conjugate transpose of a matrix with complex entries. This is perfectly consistent if one thinks of a complex number x + iy as a one-by-one complex matrix ( x + iy ). See also Exercise 1.6. The absolute value of a complex number, also known as the modulus, is denoted... 

Then define T to be the linear transformation from VP to V whose matrix in the given basis is the conjugate transpose A of A for each j and k wq have matrix entries... 

Note that on a finite-dimensional vector space V, a linear operator is Her-mitian if and only if T = T. More concretely, in C", a linear operator is Hermitian-symmetric if and only if its matrix M in the standard basis satis-lies M = M, where M denotes the conjugate transpose matrix. To check that a hnear operator is Hermitian, it suffices to check Equation 3.2 on basis vectors. Physics textbooks often contain expressions such as (+z H — z). These expressions are well defined only if H is a Hermitian operator. If H yNQK not Hermitian, the value of the expression would depend on where one applies the H. 

Exercise 3.25 Suppose M is an nyconjugate transpose of M. Suppose every eigenvalue of M is strictly positive. Define... 

Note that every matrix AT of 5 is Hermitian symmetric, i.e., writing M to denote the conjugate transpose of M, v/e have M = M. Note also that the trace of each M e 5 is zero and... 

State whether each of the following concepts is applicable to all matrices or to only square matrices (a) real matrix (b) symmetric matrix (c) diagonal matrix (d) null matrix (e) unit matrix (f) Hermitian matrix (g) orthogonal matrix (h) transpose (i) inverse (j) Hermitian conjugate (k) eigenvalues. 

The matrix obtained by taking the complex conjugate of each element of A and then forming the transpose is called the Hermitian conjugate (or conjugate transpose) of A and is symbolized by A" ... 

Here UT denotes the transpose matrix, namely if U = (uiyj) is given by its entries, then UT = (Ujyi) with rows and columns exchanged, and U = (ujyi) is the complex conjugate matrix with complex conjugate transposed entries. 

We saw that for the inverse of a matrix, A-1 A = A A 1 = 1, so for an orthogonal matrix ATA = AAT = 1, since here the transpose is the inverse. Check this out for the matrices shown. The complex analogue of an orthogonal matrix is a unitary matrix its inverse is its conjugate transpose. 

Chapter 11, matrix P describes a rotation, such as the result of a 90° pulse. In such instances matrix P is found to be unitary and to have a simple inverse, the complex conjugate transpose—(P 1)m = P m —in which rows and columns are interchanged and each i changes to — i. For example,... 

The complex conjugate A of A is the matrix formed by taking the complex conjugate of each element of A. Hie conjugate transpose A of the matrix A is formed by taking the transpose of A thus A = (A ) and... 

A unitary matrix is one whose inverse is equal to its conjugate transpose ... 

Left or right multiplication by A thus turns the matrix A into the unit matrix, and the inverse of a matrix is unique it thus follows that the inverse of A is obtained by taking the complex-conjugate transposed form, which means that the matrix is unitary. Note that the result in Eq. (2.22) is valid only if the matrix A is nonsingular. However, this will certainly be the case since... 

A matrix H which is equal to its complex-conjugate transpose, H = is called Hemitian. It follows that the diagonal elements of such a matrix are real, while corresponding off-diagonal elements form complex-conjugate pairs ... 

In order to apply the transformation to the interaction operator, we must also consider the effect of R on the column of bra-functions. This simply requires the inverse of the matrix, which, for a unitary matrix, is nothing but its complex conjugate transposed ... 

By reading dmvn the first column of coefficients (those of ri), the coefficients of ri in terms of the S s are obtained, since all quantities here are real. Similarly, the coefficients of r above give the expression for in terms of the S s. In matrix language, the inverse is equal to the (conjugate) transpose. The result is that... 

All the above considerations apply particularly to square matrices, but it is often useful to extend the concepts to rectangular arrays of elements. I he definitions of elements, equality, sum, null matrix, product, commutation, transpose, complex conjugate, conjugate, transpose, and real matrix, all apply equally well to rectangular matrices. Note that AB exists only if the number of columns of A equals the number of rows of B, whether either one is square or rectangular. Therefore, for rectangular matrices, if AB exists, BA may or may not exist. 

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